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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcALTVlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for rhmsubcALTV 42108. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rngcrescrhmALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcrescrhmALTV.c | ⊢ 𝐶 = (RngCatALTV‘𝑈) |
rngcrescrhmALTV.r | ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) |
rngcrescrhmALTV.h | ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) |
Ref | Expression |
---|---|
rhmsubcALTVlem2 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5148 | . . . 4 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → 〈𝑋, 𝑌〉 ∈ (𝑅 × 𝑅)) | |
2 | 1 | 3adant1 1079 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → 〈𝑋, 𝑌〉 ∈ (𝑅 × 𝑅)) |
3 | fvres 6207 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∈ (𝑅 × 𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) = ( RingHom ‘〈𝑋, 𝑌〉)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) = ( RingHom ‘〈𝑋, 𝑌〉)) |
5 | df-ov 6653 | . . 3 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
6 | rngcrescrhmALTV.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
7 | 6 | fveq1i 6192 | . . 3 ⊢ (𝐻‘〈𝑋, 𝑌〉) = (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) |
8 | 5, 7 | eqtri 2644 | . 2 ⊢ (𝑋𝐻𝑌) = (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) |
9 | df-ov 6653 | . 2 ⊢ (𝑋 RingHom 𝑌) = ( RingHom ‘〈𝑋, 𝑌〉) | |
10 | 4, 8, 9 | 3eqtr4g 2681 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 〈cop 4183 × cxp 5112 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 Ringcrg 18547 RingHom crh 18712 RngCatALTVcrngcALTV 41958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-res 5126 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: rhmsubcALTVlem3 42106 rhmsubcALTVlem4 42107 |
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